Let and be polynomials with the degree of the former less than the degree of the latter.
- If all complex zeroes of are simple, then
- If the different zeroes of have the multiplicities , respectively, we denote\, ;\, then
A special case of the Heaviside formula (1) is
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Example. \, Since the zeros of the binomial are\, ,\, we obtain
\\
Proof of (1). \, Without hurting the generality, we can suppose that is monic.\, Therefore
For\, ,\, denoting
one has\, .\, We have a partial fraction expansion of the form
with constants .\, According to the linearity and the formula 1 of the parent entry,
one gets
For determining the constants , multiply (3) by .\, It yields
Setting to this identity \,\, gives the value
But since\, ,\, we see that\, ;\, thus the equation (5) may be written
The values (6) in (4) produce the formula (1).
[1]
- ↑ {\sc K. V\"ais\"al\"a:} Laplace-muunnos .\, Handout Nr. 163.\quad Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).